Is |a vector+b vector| greater than |a vector| + |b vector|
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What is the value of ( A + B ) . (A × B )
We know from scalar triple product, that for vectors A, B, and C,
A . ( B × C) = B . ( C × A) = C . ( A× B) = (A×B) . C
In our question
( A + B ) . ( A × B ) = A. ( A× B) + B . ( A × B)
Now
A. ( A× B) = B. ( A×A)= 0
and
B . ( A ×B ) = A. ( B× B)= 0
Therefore,
( A +B ) . ( A × B ) = 0 +0= 0
The dot product of the sum of two Vectors with the cross products of the same two vector is zero and a scalar.
We can understand the answer as below. The sum of two vectors A and B is a vector in the plane defined by A and B. Whereas A×B is a vector which is perpendicular to the plane of vectors A and B. The dot product of two vectors one in the plane of vectors A,B and the other perpendicular to the plane is zero. Furthermore being a dot product it is a scalar.
We know from scalar triple product, that for vectors A, B, and C,
A . ( B × C) = B . ( C × A) = C . ( A× B) = (A×B) . C
In our question
( A + B ) . ( A × B ) = A. ( A× B) + B . ( A × B)
Now
A. ( A× B) = B. ( A×A)= 0
and
B . ( A ×B ) = A. ( B× B)= 0
Therefore,
( A +B ) . ( A × B ) = 0 +0= 0
The dot product of the sum of two Vectors with the cross products of the same two vector is zero and a scalar.
We can understand the answer as below. The sum of two vectors A and B is a vector in the plane defined by A and B. Whereas A×B is a vector which is perpendicular to the plane of vectors A and B. The dot product of two vectors one in the plane of vectors A,B and the other perpendicular to the plane is zero. Furthermore being a dot product it is a scalar.
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